Rank Minimization with Structured Data Patterns
The problem of finding a low rank approximation of a given measurement matrix is of key interest in computer vision. If all the elements of the measurement matrix are available, the problem can be solved using factorization. However, in the case of missing data no satisfactory solution exists. Recent approaches replace the rank term with the weaker (but convex) nuclear norm. In this paper we show that this heuristic works poorly on problems where the locations of the missing entries are highly correlated and structured which is a common situation in many applications.
Our main contribution is the derivation of a much stronger convex relaxation that takes into account not only the rank function but also the data. We propose an algorithm which uses this relaxation to solve the rank approximation problem on matrices where the given measurements can be organized into overlapping blocks without missing data. The algorithm is computationally efficient and we have applied it to several classical problems including structure from motion and linear shape basis estimation. We demonstrate on both real and synthetic data that it outperforms state-of-the-art alternatives.
KeywordsSingular Value Decomposition Rank Function Measurement Matrix Convex Relaxation Convex Envelope
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- 1.Angst, R., Zach, C., Pollefeys, M.: The generalized trace-norm and its application to structure-from-motion problems. In: International Conference on Computer Vision (2011)Google Scholar
- 2.Aquiar, P.M.Q., Stosic, M., Xavier, J.M.F.: Spectrally optimal factorization of incomplete matrices. In: IEEE Conference on Computer Vision and Pattern Recognition (2008)Google Scholar
- 5.Bregler, C., Hertzmann, A., Biermann, H.: Recovering non-rigid 3d shape from image streams. In: IEEE Conference on Computer Vision and Pattern Recognition (2000)Google Scholar
- 7.Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58(3), 11:1–11:37 (2011)Google Scholar
- 9.Garg, R., Roussos, A., de Agapito, L.: Dense variational reconstruction of non-rigid surfaces from monocular video. In: IEEE Conference on Computer Vision and Pattern Recognition (2013)Google Scholar
- 12.Jacobs, D.: Linear fitting with missing data: applications to structure-from-motion and to characterizing intensity images. In: IEEE Conference on Computer Vision and Pattern Recognition (1997)Google Scholar
- 13.Jojic, V., Saria, S., Koller, D.: Convex envelopes of complexity controlling penalties: the case against premature envelopment. In: International Conference on Artificial Intelligence and Statistics (2011)Google Scholar
- 15.Lewis, A.S.: The convex analysis of unitarily invariant matrix functions (1995)Google Scholar
- 16.Lucas, B.D., Kanade, T.: An iterative image registration technique with an application to stereo vision. In: International Joint Conference on Artificial Intelligence (1981)Google Scholar
- 18.Olsson, C., Oskarsson, M.: A convex approach to low rank matrix approximation with missing data. In: Scandinavian Conference on Image Analysis (2009)Google Scholar
- 20.Rockafellar, R.: Convex Analysis. Princeton University Press (1997)Google Scholar
- 21.Strelow, D.: General and nested Wiberg minimization. In: IEEE Conference on Computer Vision and Pattern Recognition (2012)Google Scholar
- 22.Wang, S., Liu, D., Zhang, Z.: Nonconvex relaxation approaches to robust matrix recovery. In: International Joint Conference on Artificial Intelligence (2013)Google Scholar
- 24.Zheng, Y., Liu, G., Sugimoto, S., Yan, S., Okutomi, M.: Practical low-rank matrix approximation under robust L 1-norm. In: IEEE Conference on Computer Vision and Pattern Recognition (2012)Google Scholar